improved deterministic algorithms for decremental transitive closure and strongly connected...

Improved Deterministic Algorithms for Decremental Reachability and Strongly Connected Componentsby Jakub ŁąckiA presentation by Elad

KatzAlgorithms Seminar

Instructed by Prof. Uri Zwick

Fall 2012-13Tel Aviv University

Greg’s Cable Map

What happened if Tamares Telecom (brown) went out of business?What about Med Nautilus (dark blue)?

Who got disconnected?

Problem DefinitionLet G=(V, E) be a directed graph, n=|V|, m=|E|.

query(u, v) – check if there is a directed path u → v

delete(u, v) – delete the edge (u, v)

Goal: O(1) query time, minimize deletion timeProblem: Deleting a single edge might affect possible queries!

2( )n

Solution: minimize deletion time of all edges in arbitrary order

Previous ResultsLa Poutre and van Leeuwen DeterministicFrigioni DeterministicDemetrescu and Italiano DeterministicHenzinger and King* Monte CarloBaswana Monte CarloRoditty and Zwick Las Vegas

For maintenance of strongly connected components:Frigioni** DeterministicRoditty and Zwick Las Vegas* query time ** when deleted edges are selected at random

2( )O m2( )O m

3( )O n2( log )O mn n

43 3( log )O mn n

( )O mn

2( )O m( )O mn

( )lognOn

( )O mn

Definitions

DefinitionsGu v u v if there is a directed path from to .


1 2 1 2, G

S V

s s S s s

st rongly connA non-empty set is if foreve

ecte

dry .


1 2 1 2, G

S V

s s S s s


ecte

dry .

A is aninclusio

st rongly connen-maximal st ro

cted congly co

mponnenent

cted (SCC)

set .


1 2 1 2, G

S V

s s S s s


ecte

dry .

A is aninclusio


cted congly co

mponnenent

cted (SCC)

set .( , )u v v uAn edge is an of and anin-edge out - oe ge f d .


1 2 1 2, G

S V

s s S s s


ecte

dry .

A is aninclusio


cted congly co

mponnenent

cted (SCC)


A is a gradirected ph with nacyclic graph (DAG o directed cyc) les.


1 2 1 2, G

S V

s s S s s


ecte

dry .

A is aninclusio


cted congly co

mponnenent

cted (SCC)



vA vertex in a DAG is a if it has nosource sink

in-edgesand a if it has no out -edges.


1 2 1 2, G

S V

s s S s s


ecte

dry .

A is aninclusio


cted congly co

mponnenent

cted (SCC)



vA vertex in a DAG is a if it has nosource sink

in-edgesand a if it has no out -edges.

( )n O m no isolated vert icAssumpt i es on:

FindUnreachableDown( , , )

( , )

G w S G wS V

U I U V w I E U

- DAG , source ,a set which contains all other sources.

- unreachable from , incident to .

Input :

Output :

1

2 3

4

5

6

7

8

9

:::

UIQ 6 9

6(6,7)

7

9(9,3) (9,7)7

(7, 2)(7, 8)

8

8

FindUnreachableDown - Proof

1,...,.

Let be a topological ordering of .Assume by induct ion iff it is unreachable from .If , it is inserted to at some point .If it is inserted at line 6, it is a source

k

j

i

v v Gj i v U w

v U Qw

Proof :

1

( , ) ( )

, therefore it is unreachable from .If it is inserted at line 14, it cannot be , which is a source.By the induct ion hypothesis is unreachable from for all ,hence is unreachable

j j i

i

wv

v w v v E G

v

from .If is unreachable from , it is either a sourceor all it s predecessors are unreachable from .If it is a source, it is not it self and is inserted to at line 6.Otherwise, by the induct ion

i

wv w

ww Q hypothesis all it s predecessors,

unreachable from , are inserted to .Consequent ly all it s in-edges are deleted, and it is inserted to at line 14.

w QQ

The algorithm is correct .Lemma : The algorithm is correct .Lemma :

FindUnreachableDown –Runtime Analysis

Line 5 requires t ime.Lines 8-9 require t ime, since no vertex is inserted to more than once.Lines 10, 12 require t ime, as no edge is processed more than once.Overall, The algorithm runs in

O S

O U Q

O I

O t ime.S U I

FindUnreachableDown - Usage

( )There exists a total t ime determinist ic algorithmfor decremental single-source reachability of a .

O mDAG

Lemma:



O mDAG

Lemma:

( )

( , ) , ,

( , )

We maintain the set of vert ices reachable from the source .The init ial set is obtained using BFS in .After each delet ion of we call FindUnreachable( ) ,and delete returned

sO m

u v E G v s

U I

Proof :

( ) from the graph.

The total size of all outputs of calls t o FindUnreachable is ,so we get the required running t ime.

O m



O mDAG

Lemma:

( )

( , ) , ,

( , )


sO m

u v E G v s

U I

Proof :

( ) from the graph.


O m

( )There exists an total t ime determinist ic algorithmfor decremental maintenance of the t ransit ive closure of a .

O mnDAG

Corollary:



O mDAG

Lemma:

( )

( , ) , ,

( , )


sO m

u v E G v s

U I

Proof :

( ) from the graph.


O m

( )There exists an total t ime determinist ic algorithmfor decremental maintenance of the t ransit ive closure of a .

O mnDAG

Corollary:

Run the above algorithm for every vertex.Proo f :

FindUnreachable( , , )

( , )

, ,t

G w S G wS V

U I w U VI E U

G w S

- DAG , sink ,a set which contains all other sinks.

- is unreachable from , incident to .

FindUnreachableDown( )

Input :

Output :

FindUnreachableUp

Implementation:

( , , )

( , )

, ,t

G w S G wS V

U I w U VI E U

G w S

- DAG , sink ,a set which contains all other sinks.

- is unreachable from , incident to .

FindUnreachableDown( )

Input :

Output :

FindUnreachableUp

Implementation:

( , )

( , ) G G

G S G s tS V

U I U V u U s u tI E U

- DAG with dist inguished source and sink ,a set which contains all other sources and sinks.

- , there is no path incident to .

FindUnreaInput :

Output :

chable

, , , ,G s S G t SFindUnreachableDown( ) FindUnreachableUp( )Implementation:

FindUnreachable

FindUnreachable - Examples

out

in

out

in

Decremental Maintenance of SCCs

,

,, ( , )

Let be a directed graph, , .query( ) - check if and are in the same SCCdelete( ) - delete the edge

G V E n V m E

u v u vu v u v

Decremental Maintenance of SCCs

,

,, ( , )

Let be a directed graph, , .query( ) - check if and are in the same SCCdelete( ) - delete the edge

G V E n V m E

u v u vu v u v

We can find init ial SCCs in linear t ime is st rongly connected.G

Assumption:

More Definitions

More Definitions is a directed graph obtained from by cont ract ing

all it s SCCs, preserving mult iple edges but not C

sondense

elf lo)

s.(

opGG



sondense

elf lo)

s.(

opGG

, is the graph obtained from by split t ing into vert ices and .

is given all the in-edges, is given all the out -edges. and are the source and sink used

Spl

by Find

it ( )

Unr h

eac

in out

in out

in out

G dd d

d dd d

G d

able.



sondense

elf lo)

s.(

opGG



Spl

by Find

it ( )

Unr h

eac

in out

in out

in out

G dd d

d dd d

G d

able.

:, ,S Cplit ondeAndConde nse(Splinse( )) t ( )G G dd



sondense

elf lo)

s.(

opGG



Spl

by Find

it ( )

Unr h

eac

in out

in out

in out

G dd d

d dd d

G d

able.

:, ,S Cplit ondeAndConde nse(Splinse( )) t ( )G G dd

1

1 2

2, , , is the graph obtained from byreplacing all occurrences of and withMerg )

.e( H d

d Gd d G

d d

Example1

2 3

4

5

6

7

Example1

2 3

4

5

6

7

Split

1out

2 3

4 5

6

7

1in

Split

Example1

2 3

4

5

6

7

Split

1out

2 3

4 5

6

7

1in

Split

Condense

2 3 4

7

5

6

Example1

2 3

4

5

6

7

Split

1out

1in

Split

Condense

Example1

2 3

4

5

6

7

SplitSplit

Condense

Merge

1

5

6

7

2 3 42 3 4

7

5

6

1out

1in

,( , ) ,

Let be a graph and define SplitAndCondense( ) .Let be the result returned by FindUnreachable( ) ,where contains all sources and sinks in except for and .(1) All eleme

d

d

d in out

G G G dU I G SS G d d

Le : mma

nts of Merge( ) form seperate SCCs of .U G

Back to FindUnreachable

,( , ) ,


d

d

d in out


Le : mma

nts of Merge( ) form seperate SCCs of .U G

1 2,

If there exists a path , Merge( ) .Fix , then is a st rongly connected set in .Assume by cont radict ion is not maximal, then there arevert ices s.t . there is a path

dGout ind d U U

u U u Gu

v u v u v

Proof :

1 2 1

1

2

.If the path includes , there is a path , correspondingto a path , and , a cont radict ion.If the path does not include , it corresponds to a path

d d

d

G G

G G

G Gin out

G

v v

d d v d

d u d u Ud

u v

, ...

( )

, but is a DAG, a contradict ion.If there is no path , . No cycle of contains ,

thus split t ing does not break any other SCC in ,and Merge( )=Merge( ) are the SCCs

d

d

Gd

Gout in d

d

u G

d d U V G G dd G

U V G

of .G

Back to FindUnreachable

Back to FindUnreachable (2) ,

( , ) ,Let be a graph and define SplitAndCondense( ) .

Let be the result returned by FindUnreachable( ) ,where contains all sources and sinks in except for and .(1) All eleme

d

d

d in out


Le : mma

( ) \

( )

nts of Merge( ) form seperate SCCs of .(2) The SCC of which contains is given by Merge( ) if , and otherwise.

d

d

U G

G d V G U

U V G d




d

d

d in out


Le : mma

( ) \

( )

nts of Merge( ) form seperate SCCs of .(2) The SCC of which contains is given by Merge( ) if , and otherwise.

d

d

U G

G d V G U

U V G d

( )\ ( )\

( ) \ ( ) \

( ) \

If there exists a path , .From (1) , the SCC containing is a subset of Merge( ) .It is a SCC because for every there is a path

d

d d

Gout in d

d

d

V G U V Gout

d d U V G

d V G U V G U

v V G U

d v

Proof :

, corresponding to a path .If there is no path , .

The SCC containing is , otherwise there is a path through another vertex of the SCC, a cont radic

d

d

U G Gin

Gout in d

Gout in

d d v d

d d U V G

d d d d

t ion.




d

d

d in out


Le : mma

( ) \

( )

( ) \

nts of Merge( ) form seperate SCCs of .(2) The SCC of which contains is given by Merge( ) if , and otherwise.(3) The SCCs of are Merge( ) .

d

d

d

U G

G d V G U

U V G d

G V G U U




d

d

d in out


Le : mma

( ) \

( )

( ) \


d

d

d

U G

G d V G U

U V G d

G V G U U

( ) \ ( ) \

( ) \

( ) \

If , from (1) and (2) the SCCs are given byMerge( ) Merge( ) Merge( ) .

If , is an empty set and from (1)the SCCs are given by Merge( ) Merge( ) .

d

d d

d d

d

U V G

V G U U V G U U

U V G V G U

U V G U U

Proof :

Back to FindUnreachable (4)

( , ) ( ) ' '

( , ) ', ,

Let be a st rongly connected graph and defined as above.Delete from and obtaining and .Let be the result computed by FindUnreachable( ) .(1) The SCC of

d

d d d

d

G Gu v E G G G G G

U I G u v

Corollary:

( ) \

\

containing shrinks to Merge( ) .(2) New SCCs, given by Merge( ) , emerge.

dG d V G U d

U d

,( , ) ,


d

d

d in out


Le : mma

( ) \

( )

( ) \


d

d

d

U G

G d V G U

U V G d

G V G U U


( , ) ( ) ' '

( , ) ', ,

Let be a st rongly connected graph and defined as above.Delete from and obtaining and .Let be the result computed by FindUnreachable( ) .(1) The SCC of

d

d d d

d

G Gu v E G G G G G

U I G u v

Corollary:

( ) \

\

containing shrinks to Merge( ) .(2) New SCCs, given by Merge( ) , emerge.

dG d V G U d

U d

,( , ) ,


d

d

d in out


Le : mma

( ) \

( )

( ) \


d

d

d

U G

G d V G U

U V G d

G V G U U

Direct ly from the lemma.Proof :


' ( , )

' ( ', ') | ' | | ' |

Let , , and be defined as in the previous corollary.Assuming we have already computed , We can obtain an implicitrepresentat ion of Condense( ) in t ime.

d

d

C C C C

G G G U IG

G V E O V E

Lemma:


' ( , )

' ( ', ') | ' | | ' |


d

d

C C C C

G G G U IG

G V E O V E

Lemma:

'From the corollary, we can compute the vertex set of

Condense( ) using FindUnreachable. It s edges can be obtainedfrom by replacing every edge endpoint which does not belongto with , the vd

GI

U v

Proof :

ertex of Condense( ) containing .G d


' ( , )

' ( ', ') | ' | | ' |


d

d

C C C C

G G G U IG

G V E O V E

Lemma:



GI

U v

Proof :


dG G When deleting edges from , we can track strong connectivity of !


' ( , )

' ( ', ') | ' | | ' |


d

d

C C C C

G G G U IG

G V E O V E

Lemma:



GI

U v

Proof :



What about other Q: edges?


' ( , )

' ( ', ') | ' | | ' |


d

d

C C C C

G G G U IG

G V E O V E

Lemma:



GI

U v

Proof :



What about other Q: edges?, Maintain SCCs of Split ( ) - A: G d Trees!

Even More Definitions


,

,,

Let be a st rongly connected graph. An of is:If , a single node containing .Otherwise, a root containing SplitAndCondense( ) ,and a child for each SCC SplitAndCo

SCC

nde

-t r

nse( )which i

ee G G

G v v

G dv G d

s the SCC-t ree of , except for only one child

for both and .in out

vd d


,

,,


SCC

nde

-t r

nse( )which i

ee G G

G v v

G dv G d



vd d

A is a vertex of a SCC-tnode ree.


,

,,


SCC

nde

-t r

nse( )which i

ee G G

G v v

G dv G d



vd d

A is a vertex of a SCC-tnode ree.An is a non-linne eaf r node node.


,

,,


SCC

nde

-t r

nse( )which i

ee G G

G v v

G dv G d



vd d


( )For any node :

is it s parent node.N

p N


,

,,


SCC

nde

-t r

nse( )which i

ee G G

G v v

G dv G d



vd d


( )For any node :

is it s parent node.N

p N

( ) is the graph in the node.D N

Example1

2 3

4

5

1in 2 3 4 5 1out

2in 2out34 5 1

2 34in 5 4out

4 5

Constructing SCC Trees

( )( )

An SCC-t ree can be const ructed in t ime,where is the depth of the t ree. It requires space.

O mO n m

Lemma:

Constructing SCC Trees

( )( )

An SCC-t ree can be const ructed in t ime,where is the depth of the t ree. It requires space.

O mO n m

Lemma:

( )( )

The t ree can be const ructed in t ime using a linear t imealgorithm for finding SCCs, as each level of the t ree takes t ime.

O mO m

Proof :

( )The t ree has leaves and each inner node has at least two children,therefore the total number of nodes is .There is a biject ion between vert ices in graphs of inner nodesto their children in the

nO nD

( ) t ree, hence the total number of vert ices

in all graphs is . Also, each edge is stored in only one node.Consequent ly we get the required space.

D O n We do not store the SCCs explicitly!

Deleting an Edge 1

2 3

4

5

1in 2 3 4 5 1out

2in 2out3 1

2 34 5

4

5

1in2 3 4 1out

2in 2out3 1

2 34

4

5

5

2in 2out3 1

2 34

4 5

, : [ ] [ ]query( )u v SCC u SCC v

1in 2 3 4 5 1out

2in 2out34 5 1

2 34in 5 4out

4 5

Runtime Analysis( )

(1) ( )

If the depth of an SCC-t ree is , the algorithm processesany sequence of edge delet ions in total t ime and answerseach query in t ime, using space.

O mO O n m

Theor em:

Runtime Analysis( )

(1) ( )


O mO O n m

Theor em:

( )( )

( ) ( )

By the lemma const ruct ing the SCC-t ree takes t imeand uses space.Every edge is lifted up the t ree t imes, a total of lift -ups.

FindUnreachable takes linear t ime with

O mO n m

O O m

Line 1:

Proof :

respect to it s input and output .It s output consists of edges to be lifted up or deleted,and it s input consists of vert ices lifted-up,which gives the required running t ime.

Each subt ree is liftLine 4 : ( )( )

( )( )

ed up t imes. Each vertex and edge are lifted up t imes.Each vertex incident to is lifted up t imes.Each subt ree is lifted up t imes.

OO

e OO

Line 6 :Line 8 : Line 10 :

( )(1) ( )


O mO O n m

Theor em:

Runtime Analysis (2)

( )(1) ( )


O mO O n m

Theor em:


( )(1)

( )

There exists an total t ime determinist ic algorithmfor decremental maintenance of SCCs which answers queries in t imeand uses space.

O mnO

O n m

Corollary:

( )(1) ( )


O mO O n m

Theor em:


( )(1)

( )


O mnO

O n m

Corollary:

( )Direct ly From the theorem, we have proved .O n Proof :

( )(1) ( )


O mO O n m

Theor em:


( )(1)

( )


O mnO

O n m

Corollary:

( )Direct ly From the theorem, we have proven .O n Proof :

What about the transitive closure?

Back to the Transitive Closure

( )

(1) ( )

There exists an total t ime determinist ic algorithmfor decremental maintenance of the t ransit ive closure which answersqueries in t ime and uses space.

O mn

O O n m

Theorem:

Back to the Transitive Closure

( )

(1) ( )

There exists an total t ime determinist ic algorithmfor decremental maintenance of the t ransit ive closure which answersqueries in t ime and uses space.

O mn

O O n m

Theorem:

We first build SCC-t rees for each SCC of the graph,then the SCCs are cont racted to create a DAG.When delet ing an edge from the DAG we use the algorithmfrom the first corollary.When delet ing an

Proof :

edge from inside a SCC we use the algorithm from thelast corollary, if the SCC decomposes we use FindUnreachableDownwith the set of vert ices of the decomposed SCC as the second parameter.When checking [ ] [ ] for a path from to we check if or if there is a path from the SCC of t o the SCC of .

u v SCC u SCC vu v

Planar Graphs( ) 8

0 1A planar graph , , has a subset of vert ices whose

removal decomposes it into components of size at most easepe

ch, .The subset can be found in linear t imrator e.

G V G n nn

Theorem*:

* Lipton & Tarjan, 1979

Planar Graphs( ) 8




G V G n nn

Theorem*:

1.5

( , )

( )

Let be a directed planar graph and .There exists an algorithm for decremental maintenance of it s SCCswhich runs in total t ime.

G V E V n

O n

Theorem:

* Lipton & Tarjan, 1979

Planar Graphs( ) 8




G V G n nn

Theorem*:

1.5

( , )

( )

Let be a directed planar graph and .There exists an algorithm for decremental maintenance of it s SCCswhich runs in total t ime.

G V E V n

O n

Theorem:

2

( )(1) ( ) (1)

( ) 8 ( ) 8 8 ( )

We use the previous algorithm for maintaining SCCs,only choosing vert ices to split from the seperator.The depth of the SCC-t ree is then bounded:nn O n O

n n n n n n

P of ro :

1.5

8... (1)1

( ) ( )Therefore, the algorithm runs in overall t ime.

n O O n

O m O n

* Lipton & Tarjan, 1979 3 6 ( ) In planar graphs .m n O n Note:

The End

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